Equilibrium selection via current sheet relaxation and guide field amplification

Although there is a continuous spectrum of current sheet equilibria, how a particular equilibrium is selected by a given system remains a mystery. Yet, only a limited number of equilibrium solutions are used for analyses of magnetized plasma phenomena. Here we present the exact process of equilibrium selection, by analyzing the relaxation process of a disequilibrated current sheet under a finite guide field. It is shown via phase-space analyses and particle-in-cell simulations that the current sheet relaxes in such a way that the guide field is locally amplified, yielding a mixed equilibrium from the spectrum. Comparisons to spacecraft observations and solar wind current sheet statistics demonstrate that such mixed equilibria are ubiquitous and exist as underlying local structures in various physical environments.


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After 10 years in academia, Bart originally joined Nature Physics in 2013, where he handled topics from condensed-matter physics and plasma physics. He also launched a column called 'Measure for Measure' on aspects of metrology. In July 2017, Bart moved to Berlin to become Regional Executive Editor for the newly established Nature Portfolio office in Germany, which he helped develop as an editorial and publishing hub with a focus on local outreach. Since 2021, he combines that capacity with the role of Senior Consulting Editor at Nature Physics, where he covers soft-condensed-matter physics and has a consulting role for Nature Communications.

Editorial assessment and review synthesis Editor's summary and assessment
The authors look at current sheets in plasmas (space and laboratory)these are regions in plasmas that are in between two regions of opposing magnetic field and store magnetic free energy. By means of analytical theory and particle-in-cell simulations, the authors here present insights on what makes these sheets be in equilibrium, or how they relax into equilibrium states.

Editorial synthesis of reviewer reports
The 2 referees are generally positive about the work, but have suggestions for improvements. Referee #2 questions the claim of universality, however.

Nature Physics
Revision not invited Major revisions with extension of the work Major revisions Minor revisions Revisions not needed The work is of interest to the specialists from the (space) plasma community, but unfortunately, not of sufficiently broad relevance to be suitable for publication in Nature Physics.

Nature Communications
Revision not invited Major revisions with extension of the work Major revisions Minor revisions Revisions not needed If the authors can make a convincing case that the results are sufficiently universally valid, then Nature Communications could be an appropriate venue for publication.

Communications Physics
Revision not invited Major revisions with extension of the work Major revisions Minor revisions Revisions not needed If it turns out that the results are not sufficiently universally applicable/valid, then Communications Physics would be the best fit.

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Annotated reviewer reports
The editors have included some additional comments on specific points raised by the reviewers below, to clarify requirements for publication in the recommended journal(s). However, please note that all points should be addressed in a revision, even if an editor has not specifically commented on them.

Reviewer #1 information Expertise
Space plasmas, magnetic reconnection

Editor's comments
The referee is positive about the work reported but has some comments that required addressing in a revised version.

Reviewer #1 comments Section Annotated Reviewer Comments
Remarks to the Author: Overall significance This paper presents a possible process by which to determine the equilibrium outcome of the relaxation of a disequilibrated current sheet in the presence of a small finite guide field. This is an extension of their previous work of the same process without a guide field being present. It is my opinion that this work will provide a basis and framework for future work in current sheet behavior. Although it need not be included in this work, additional study should be conducted with higher current sheet aligned energies and with higher mass species. I feel that this paper should be published after addressing of several minor comments.
The referee appreciates the work, is supportive of publication, but lists a series of issues that need to be addressed.
General Comment.
Too many unneeded modifiers. These words obsfucate the message of the paper. This makes it incredibly more difficult to read. Words like "are exhaustively classified" should be "are classified". "in order to fathom the equilibrium" should be "in order to understand the equilibrium" fathom is more than a little 'over the top' Specific Comments. Was this study run in 2D or 3D? I assume 2D since you made "out of plane" streak plots, but this is not explicitly stated.
FIG 1 is not clear enough to make sense. Consider more traditional line plots of B and rho as cuts in the YX plane. The arrow widths are nearly impossible to decipher and the pink on pink color choice was just bad. What does it mean when the lines are closer or further apart in d)? Explain this to the reader if you maintain these plots.
In the text body below Figure 1 you explain "Fig 1d exhibits  you specified that X is normal to the CS.
Am I correct in assuming that: 1) Z direction is along the CS.
2) Y is across the CS.
3) that your XYZ would be ZXY in GSE coordinates (tail).
As such Column 2 (d-e-f) shows movement along the CS and Column 3 (g-h-i) across the CS? Am I correct in assuming that particle motion is along the CS (by some means) and across the CS due to the Guide Field?
Particles can move in both + and -directions along the current sheet and across the current sheet.
In FIG 2 what decides whether the red, blue or cyan will move: one way or the other (+/-Z) along the CS (d-e-f)?
In FIG 2 what decides whether the red, blue or cyan will move: one way or the other (+/-Y) across the CS (g-h-i)?
I see that later you specify subclasses based on the sign of -- Figure 2 The green and magenta colors represent By. I assume the colors are plus and minus but it is not stated. -I retract this initial comment and replace it with: I suggest that you move the By and Bz arrows above the column or give them a white background. Pink on pink and Green on green is very hard to see.
What is your XYZ coordinate system? It does not seem to be any standard system such as GSE. Also you are comparing XYZ with LMN which is given in reference to GSE.
This produces confusion.
Please clarify how the LMN coordinate system relates to your XYZ system? I mean in general terms like 'along the CS', 'perpendicular to the CS', 'across the CS'.

Editor's comments
The referee finds the work of importance, interest and impact, but questions the universality of the results as claimed by the authors.

Remarks to the Author: Overall significance
This paper concerns the problem of relaxation of an out of equilibrium current sheet in collision-less plasmas. The authors are interested in describing how the new equilibrium is selected. They present a theoretical derivation of particle orbits, particle-in-cell numerical simulations, and observations. I don't have any major concerns about the quality of the work or how the paper is currently written.

Remarks to the Author: Impact
The subject of the paper is of great importance and potential impact.

Remarks to the Author: Strength of the claims
The claim of "universality" that the author makes doesn't seem to be supported enough by the analysis the authors present. In other words, the authors claim that the process they have studied is universal, but the paper does not contain convincing proof of such a statement. For this reason, although the scientific quality of the work is excellent, I don't think that its current version meets the journal requirements.
From an editorial point of view, the validity of the universality claim is important in deciding the best journal to publish this work in. If a strong case can be made that there is indeed universality in the results, Nature Communications could be considered as the best venue for the paper. If not, then Communications Physics would be more suitable.
In what follows, I list those points that, in my opinion, should be improved to arrive at a more convincing version of this study.
1. As it is commonly done in PIC simulations, the authors use a nonrealistic speed of light to Alfvén speed ratio (c/va). The results presented in the Extended Data section show that a larger value of c/va results in the production of waves. The author state that these plasma oscillations damp away without affecting the core relaxation mechanisms. I have a few questions on this point. Which kind of waves are these? Why do they develop when a larger c/va is considered? What dumps these waves? What is the effect of periodicity on the evolution of these waves inside the system? In an open system, would such waves leave the current sheets? Are these waves observed by MMS or any other satellite?
2. The simulations the authors have performed are 1D. However, the stability of a single current sheet may depend on 3D dynamics. What justifies the 1D approach the authors use? In their conclusions, the authors also admit that in a 2D case they have studied, reconnection spontaneously develops, which makes questionable the existence in natural environments of a 1D equilibrium like what they are presenting in the paper.
3. Could the authors better explain why the perpendicular electron temperature in their simulation is everywhere larger than the parallel one (Fig 6 panel e-f) while it is the opposite in the observed current sheet? They make an argument on that just before the section "Discussion and outlook"; however, it was hard for me to understand what they mean.
4. The authors show that their numerical simulations are in accordance with their analysis of particle trajectories. Would it be possible for them to analyze MMS data and extract the VDF for ions and electrons? How do these compare with those obtained from numerical simulations and the analysis of particle orbits?
5. At the beginning of page 14, the authors note a discrepancy between the analytical and the numerical solution. This discrepancy seems to be a weak point of the paper since the numerical solution does not relax to one of the expected equilibria. Still, it relaxes to some other equilibrium whose analytical form is unknown. It would be a great result if the authors could find an analytical form corresponding to the final state of their simulations.
6. In their final discussion, the authors claim that mixed equilibria are likely to be ubiquitous. However, they also admit that their MMS observations concern a current sheet out of equilibrium, and this seems an apparent contradiction. Are the type of equilibria discussed by the authors also frequently observed by MMS?
These remarks should help to arrive at an improved version of the apper.
Minor comment.
In Remarks to the Author: Reproducibility I don't see any significant issue in reproducing the analysis presented in the paper.

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